Practice Test

1) A vector which is collinear / coincident / parallel with any given vector is

2) Two vectors are collinear / coincident / parallel if each one of them is

3) [ i, j, k] =

4) If vectors i + j + k, i - j + k and 2i + 3j + mk are coplanar, then m =

5) If the origin O and the points A(1, 2, 3), B(2, 3, 4) and, P(x,y,z) are coplanar then

6) i . ( j x k ) + j . ( k x i ) + k . ( i x j ) =

7) If vectors 2i - j + k, i + 2j - 3k and 3i + mj + 5k are coplanar, then m is a root of the equation

8) If vectors 2i - j + k, i + 2j - 3k and 3i + aj + 5k are coplanar then a =

9) Vectors i + j + (m + 1) k, i + j + mk and i - j + mk are coplanar for

10) Which of the following is not equal to any of the remaining three ?

11) 2i.(j x k) - 3j.(i x k) - 4k.(i x j) =

12) If c is the mid-point of AB, and P is a point outside AB, then

13) If vectors ai + j + k, i - bj + k, i + j - ck are co-planar, then abc + 2 =

14) If the four points A(2-x, 2, 2), B(2, 2-y, 2), C(2, 2, 2-z) and D(1, 1, 1) are co-planar, then

15) If the vectors i + j, i - j and li + mj+ nk are coplanar, then:

16) Points A(4, 5, 1), B(0, -1, 1), C(3, 9, 4) and D(-4, 4, 4) are

17) A unit vector which is coplanar with (i + j + 2k) and ( i + 2j + k) and perpendicular to (i + j + k) is

18) If direction of ratios of two lines are 2, -6, -3 and 4, 3, -1 then direction ratios of a line perpendicular to both of them are

19) If a and b are two collinear vectors, then which of the following are incorrect ?

20) The points A (1,2,7), B (2,6,3) and C (3,10,-1) are

21) If a = b + c , then which of the following statements is correct ?

22) Find the position vector of point R which divides the line joining two points P (2a + b) and Q (a - 3b) externally in the ratio 1 : 2. Also show that P is the middle point of the line segment RQ.

23) If the points (-1, -1, 2) , (2, m, 5) and (3, 11, 6) are collinear, then the value of m is

24) If a and b are the position vectors of A and B, respectively, then the position vector of a point C in BA produced such that BC = 1.5 BA, is

25) If a + b + c = 0, then which of the following is correct ?

26) The value of [a - b b - c c - a] is equal to

27) For any three vectors a, b and c the value of [a + b b + c c + a] is equal to

28) If a and b are two non-collinear vectors and xa + yb = 0

29) Five points given by A, B, C, D and E are in a plane. Three forces AC, AD and AE act at A and three forces CB, DB, EB act at B. Then their resultant is

30) If ABCDEF is regular hexagon, then AB + EB + FC is equal to

31) If position vector of a point A is
a + 2b and any point P(a) divides AB in the ratio of 2 : 3, then position vector of B is

32) a, b and c are mutually perpendicular unit vectors, then | a + b + c | is equal to

33) If A, B, C, D and E are five coplanar points, then DA + DB + DC + AE + BE + CE is equal to

34) The moment about the point M (-2, 4, -6 ) of the force represented in magnitude and position AB, where the points A and B have the coordinators (1, 2, -3) and ( 3, -4, 2 ) respectively is

35) If x . a = x . b = x . c = 0, where x is non-zero vector.
Then, [ a x b b x c c x a ] is equal to

36) a x [ a x ( a x b )] is equal to

37) [ b x c c x a a x b ] is equal to

38) If a, b and c are the three vectors mutually perpendicular to each other to form a right handed system and |a| = 1, |b| = 3 and |c| = 5, then [ a - 2b b - 3c c - 4a ] is equal to

39) If a, b and c are unit coplanar vectors, then [ 2a - b 2b - c 2c - a ] is equal to

40) For any three vectors a, b and c, (a - b) . (b + c) x ( c + a) is equal to

41) What is the value of
(d + a) . [ a x { b x ( c x d )}] ?

42) A tetrahedron has vertical at
O (0, 0), A ( 1, 2, 1), B (2, 1, 3) and
C ( -1, 1, 2). Then, the angle between the faces OAB and ABC will be

43) The sum of two unit vectors is a unit vector. The magnitude of their difference is

44) For non-zero vector a and b, if
| a + b | < | a - b |, then a and b are

45) If ( a x b ) x c = -5a + 4b and a . b = 3, then a x ( b x c ) is equal to

46) If a and b are two unit vectors such that a + 2b and 5a - 4b are perpendicular to each other, then the angle between a and b is

47) If a + b + c = 0 and |a| = 3, |b| = 5 and |c| = 7 , then the angle between a and b is

48) The vector ( a + 3b ) is perpendicular to ( 7a - 5b ) and ( a- 4b ) is perpendicular to ( 7a - 2b ). The angle between a and b is

49) If a and b are unit vectors, then the greatest value of |a + b| + |a - b| is

50) Let u, v and w be three vectors such that |u| = 1, |v| = 2, |w| = 3. If the projection of v along u is equal to that of w along u, v and w are perpendicular to each other, then | u - v + w | is equal to

51) Vectors a and b are such that |a| = 1, |b| = 4 and a . b = 2. If c = 2a x b - 3b, then the angle between b and c is

52) If a is a unit vector and projection of x along a is 2 and a x r + b = r, then r is equal to

53) If a + 2b + 3c = 0, then
a x b + b x c + c x a = ka x b ,
where k is equal to

54) a and b are two given vectors. With these vectors as adjacent sides, a parallelogram is constructed. The vector which is the altitude of the parallelogram and which is perpendicular to a is

55) Let a and b be two non - zero perpendicular vectors. A vector r satisfying the equation r x b = a can be

56) Vector x is

57) Vector y is

58) Vector z is

59) The position vector of P is

60) The volume of the tetrahedron ABCF is

61) Let r be a non-zero vector satisfying r . a = r . b = r . c = 0 for given non-zero vectors a, b and c.
Statement I [ a - b b - c c - a ] = 0
Statement II [ abc ] = 0

62) The vectors a and b are not perpendicular c and d are two vectors satisfying b x c = b x d and a . d =0.
Then the vectors d is equal to

63) Let a, b and c be three non-zero vectors which are pairwise non-collinear. If a + 3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is

64) Which of the following expressions are meaningful?

65) The volume of the tetrahedron formed by the points (1, 1, 1), (2, 1, 3), (3, 2, 2) and (3, 3, 4) in cubic units is:

66) The summation of two unit vectors is a third unit vector, then the modulus of the difference of the unit vectors is

67) Which one of the following is not correct?

68) I. Two non-zero. Non-collinear vectors are linearly independent.
II. Any three coplanar vectors are linearly dependent.
Which of the above statements is/are true?

69) The vector equation of the sphere whose center is the point (1, 0, 1) and radius is 4, is